Deligne categories are tensor categories, introduced by P. Deligne, which provide a formal way to interpolate representation-theoretic structures attached to classical groups and supergroups (such S_n, GL(n),Sp(2n),O(n),GL(n|m),OSp(n|2m),etc.) to complex values of the integer “rank parameter” n. I will first review them and then explain how to use them to construct and study deformed double current algebras which have recently become increasingly popular in the mathematics and physics literature. This is joint work with D. Kalinov and E. Rains.